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ADDITIONAL MATHEMATICS 0606/21

Paper 2 October/November 2020

2 hours

You must answer on the question paper.

No additional materials are needed.

INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● You should use a calculator where appropriate. ● You must show all necessary working clearly; no marks will be given for unsupported answers from a

calculator. ● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in

degrees, unless a different level of accuracy is specified in the question.

INFORMATION ● The total mark for this paper is 80. ● The number of marks for each question or part question is shown in brackets [ ].

*8320369122*

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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax bx c 02 + + = ,

x ab b ac

242!

=- -

Binomial Theorem

( )a b a a b a b a bn n n

r b1 2n n n n n r r n1 2 2 f f+ = + + + ++ +- - -e e eo o o

where n is a positive integer and ( ) ! !

!nr n r r

n=-

e o

Arithmetic series ( )u a n d1n = + -

( ) { ( ) }S n a n a n dl21

21 2 1n = + = + -

Geometric series u arnn 1= -

( )

( )S ra r

r11

1n

n!=

--

( )S ra r1 11=-3

2. TRIGONOMETRY

Identities

sin cosA A 12 2+ =sec tanA A12 2= +eccos cotA A12 2= +

Formulae for ∆ABC

sin sin sinAa

Bb

Cc

= =

cosa b c bc A22 2 2= + -

sinbc A21T =

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1 Solve the inequality x x3 2 82+ + . [3]

2 Find the coordinates of the points of intersection of the curve x xy 92 + = and the line y x32 2= - .

[5]

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3 Write g gl lx y3 2+ - as a single logarithm. [3]

4 It is given that ( )ln sin cosy x x3= + for x0 21 1 r .

(a) Find xy

dd

. [3]

(b) Find the value of x for which xy

21

dd=- . [3]

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5 The first three terms in the expansion of ( )a bx x15

+ +` j are x cx32 208 2- + . Find the value of each of the integers a, b and c. [7]

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6 DO NOT USE A CALCULATOR IN THIS QUESTION.

In this question all lengths are in centimetres.

3 1+

15°

A

C

B

3 1-

In the diagram above AC 3 1= - , AB 3 1= + , angle °ABC 15= and angle °CAB 90= .

(a) Show that °tan15 2 3= - . [3]

(b) Find the exact length of BC. [2]

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7 DO NOT USE A CALCULATOR IN THIS QUESTION.

( )p x x x x2 3 23 123 2= - +-

(a) Find the value of p 21b l. [1]

(b) Write ( )p x as the product of three linear factors and hence solve ( )p x 0= . [5]

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8 The population P, in millions, of a country is given by P A bt#= , where t is the number of years after January 2000 and A and b are constants. In January 2010 the population was 40 million and had increased to 45 million by January 2013.

(a) Show that .b 1 04= to 2 decimal places and find A to the nearest integer. [4]

(b) Find the population in January 2020, giving your answer to the nearest million. [1]

(c) In January of which year will the population be over 100 million for the first time? [3]

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9 A particle moves in a straight line such that, t seconds after passing a fixed point O, its displacement from O is s m, where e es t10 12 9t t2= - - + .

(a) Find expressions for the velocity and acceleration at time t. [3]

(b) Find the time when the particle is instantaneously at rest. [3]

(c) Find the acceleration at this time. [2]

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10 The gradient of the normal to a curve at the point (x, y) is given by xx

1+ .

(a) Given that the curve passes through the point (1, 4), show that its equation is lny x x5= - - . [5]

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(b) Find, in the form y mx c= + , the equation of the tangent to the curve at the point where x 3= . [3]

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11 The equation of a curve is y x x16 2= - for x0 4G G .

(a) Find the exact coordinates of the stationary point of the curve. [6]

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(b) Find ddx x16 2 2

3

-` j and hence evaluate the area enclosed by the curve y x x16 2= - and the

lines ,y x0 1= = and x 3= . [5]

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12

4 cm

BA 5 cm

3 cm

C

D

The diagram shows a shape consisting of two circles of radius 3 cm and 4 cm with centres A and B which are 5 cm apart. The circles intersect at C and D as shown. The lines AC and BC are tangents to the circles, centres B and A respectively. Find

(a) the angle CAB in radians, [2]

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(b) the perimeter of the whole shape, [4]

(c) the area of the whole shape. [4]

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